Thus, our charges generate the action of a \({\rm w}_{1+\infty }\)  symmetry.It would be interesting to derive the commutation relation (REF ) directly from the \({\rm w}^p {\rm w}^q\)  OPE by taking an appropriate set of contour integrals. Also, there may exist a redefinition of the modes, such as the one in [1]}, under which the action (REF ) simplifies to a familiar representation of \({\rm w}_{1+\infty }\) . A proof of (REF ) is provided in Appendix REF . The proof for general \(p\)  and \(m\)  is performed by induction, using \(\widehat{\rm w}^{2}_k\)  to vary \(m\)  and \(\widehat{\rm w}^{\frac{5}{2}}_k\)  to raise \(p\) .The structure of the proof is consistent with the following observations made in [2]}, [1]}. First, \(\widehat{\rm w}^2_k\)  generates an \({\rm SL}(2, \mathbb {R})\)  subalgebra under which other \(\widehat{\rm w}^q_n\)  transform covariantly like the modes of an \({\rm SL}(2, \mathbb {R})\)  primary of weight \(q\) . Second, the modes with \(p \le 2\)  form a closed subalgebra, while \(\widehat{\rm w}^{\frac{5}{2}}_k\)  generates all higher-weight modes with \(p > \frac{5}{2}\) .
